Hedwig Kiesler was an inventor, and she might have been a great one. Scratch that: She was a great one, since the sole invention she patented during her lifetime now makes possible your GPS, cellphone, Wi-Fi technology, and Bluetooth.

But Kiesler had a curse: She was a woman and she was very, very beautiful, and so she became a Hollywood star instead. “Bombshell: The Hedy Lamarr Story” is a new documentary from writer-director Alexandra Dean that fascinates with its biographical zigs and zags — the stranger and stranger places this woman went — and ultimately stands as a tragedy of what might have been and never came to be, for both its subject and for us.

A co-production of American Masters Pictures and Reframed Pictures (it will air on PBS in May), “Bombshell” sticks to a tried and true biodoc template: archival digs, film clips, talking heads of friends and family. Dean’s ace in the hole is former Forbes journalist Fleming Meeks, who in 1990 interviewed Lamarr, then 75 and a recluse, on a tip that during World War II the actress had worked out a technology called “frequency hopping” that allowed for secure wireless communications. The tape of Meeks’s interview plays throughout Dean’s documentary like a third-act reveal in a studio thriller.

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When Meeks’ article came out, it occasioned a shamefaced cultural reappraisal: A woman lauded throughout her life as the most beautiful human being in the movies turned out to have had a brain, a technical bent, and immense curiosity. “Bombshell” assembles a portrait of Kiesler’s childhood in post-WWI Vienna as open-minded and indulgent, with a banker father who encouraged the girl’s inquiries into how things worked. She took music boxes apart and reassembled them at the age of 5; in school she excelled at chemistry. Today she’d be recognized as a STEM candidate from the get-go.

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But if you were young and pretty in bohemian Vienna, it was easy to be looked at in photos and plays and movies. One of the last, 1933’s “Ecstasy,” became a global scandal and a banned must-see for its scenes of the 18-year-old Kiesler swimming in the nude and feigning orgasm in close-up. After fleeing Austria for Paris ahead of Hitler’s Anschluss in 1937, she met MGM head Louis B. Mayer, who brought her to California, rechristened her Hedy Lamarr, and offered her up to dazzled moviegoers on the half-shell of the 1938 classic “Algiers.”

In truth — and “Bombshell” slides rather too tactfully around this and other points — Lamarr was more of a stunning onscreen presence than a gifted actress, and Mayer’s hopes that she might become his new Garbo never panned out. Her beauty was so “perfect” as to seem almost unreal — to preclude individuality — and her performances in movies like “White Cargo,” “Tortilla Flat,” and “Dishonored Lady” didn’t have the charisma to pick up the slack.

The woman behind the celebrated face was homesick in Hollywood, encouraged in her tinkering by sometime-boyfriend Howard Hughes, who consulted with her on aircraft designs. Anxious to help the American war effort, Lamarr and avant-garde composer George Anthiel devised the frequency-hopping technology to better guide radio-controlled torpedoes to their targets; they patented their invention in 1942, but it was never used during WWII.

Much of the later passages in “Bombshell” trace how Lamarr’s breakthrough caught on in Navy communications research during the late 1950s and early 1960s — US warships churning across the Caribbean during the Cuban Missile Crisis of 1962 used frequency hopping to shield their radio conversations. By the time the actress learned of this, in the 1990s, the patent had long since expired, as had her right to sue. She made no money from her development of a technology that today, in the film’s estimate, has a market value of $30 billion.

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Instead, she lived out the life of a star who is famous for her looks and has the misfortune to get old. The 1949 Biblical epic “Samson and Delilah” was Lamarr’s last big hit; her later years — attested to by two grown children and such friends as the late Turner Classic Movies host Robert Osborne (in his final interview before his death last year) — were a roller coaster of face work, “vitamin” addictions, revolving-door husbands, and the occasional arrest for shoplifting. Even so, Lamarr’s mind never stopped ticking: Her plastic surgeons recalled with awe her revolutionary suggestions on ways to improve their art.

She ended as a pop-culture joke, spoofed on TV by Lucille Ball and parodied as “Hedley Lamarr” in Mel Brooks’s “Blazing Saddles,” but she could have been something else entirely — that’s the bass-note sorrow of Dean’s documentary. Meeks’s Forbes article did bring Lamarr welcome recognition for her invention before her death in January 2000. The penultimate moments of “Bombshell” are moving, re-creating the lost Vienna of Kiesler’s childhood and overlaying the voice of the aging Lamarr, interviewed by an Austrian news team in 1970, as she speaks of never being understood in America. Adrift in the Land of the Lotus Eaters, she spent a lifetime being looked at and never once being seen.

★★★BOMBSHELL: The Hedy Lamarr Story

Written and directed by Alexandra Dean. At Kendall Square. 90 minutes. Unrated.